0.207 0.919 0.720 0.052 0.542 0.587 0.024 0.588 0.912 0.323 0.381 0.365 0.867 0.981 0.987

0.465 0.286 0.210 0.381 0.145 0.188 0.852 0.535 0.723 0.047 0.168 0.993 0.567 0.084 0.830

0.037 0.899 0.193 0.756 0.239 0.533 0.197 0.141 0.441 0.563 0.513 0.220 0.684 0.977 0.293

0.907 0.210 0.484 0.054 0.307 0.651 0.603 0.892 0.011 0.186 0.516 0.538 0.547 0.802 0.189

0.876 0.385 0.902 0.724 0.038 0.791 0.592 0.250 0.954 0.799 0.634 0.845 0.455 0.913 0.077

0.649 0.420 0.313 0.099 0.478 0.842 0.654 0.774 0.627 0.038 0.196 0.984 0.331 0.094 0.555

0.142 0.243 0.852 0.742 0.809 0.720 0.126 0.359 0.809 0.200 0.842 0.245 0.063 0.726 0.398

0.879 0.048 0.734 0.605 0.685 0.613 0.529 0.817 0.813 0.746 0.879 0.737 0.227 0.443 0.634

0.765 0.138 0.236 0.948 0.633 0.097 0.354 0.909 0.084 0.871 0.002 0.507 0.567 0.702 0.206

0.143 0.863 0.731 0.555 0.917 0.552 0.249 0.047 0.108 0.256 0.028 0.632 0.723 0.845 0.360

0.137 0.118 0.979 0.319 0.458 0.742 0.205 0.329 0.533 0.852 0.251 0.936 0.065 0.166 0.812

0.223 0.410 0.878 0.946 0.956 0.301 0.297 0.605 0.788 0.196 0.939 0.185 0.396 0.172 0.954

0.735 0.201 0.169 0.265 0.424 0.083 0.162 0.265 0.639 0.761 0.584 0.430 0.826 0.564 0.692

0.349 0.845 0.462 0.843 0.520 0.630 0.735 0.578 0.975 0.411 0.801 0.196 0.232 0.703 0.387

0.229 0.673 0.815 0.986 0.146 0.061 0.774 0.830 0.324 0.339 0.696 0.170 0.998 0.155 0.680

0.133 0.731 0.363 0.723 0.055 0.623 0.016 0.103 0.959 0.307 0.843 0.187 0.307 0.691 0.932

0.262 0.911 0.906 0.743 0.489 0.081 0.745 0.026 0.891 0.638 0.822 0.235 0.539 0.116 0.011

0.336 0.252 0.700 0.009 0.533 0.159 0.602 0.516 0.809 0.496 0.703 0.374 0.708 0.376 0.791

0.958 0.075 0.069 0.337 0.659 0.506 0.889 0.890 0.546 0.419 0.141 0.971 0.928 0.604 0.785

0.521 0.159 0.308 0.293 0.134 0.235 0.910 0.669 0.984 0.688 0.107 0.738 0.705 0.496 0.661

0.447 0.196 0.023 0.176 0.600 0.977 0.889 0.968 0.016 0.064 0.016 0.576 0.945 0.597 0.314

0.166 0.171 0.881 0.158 0.583 0.320 0.332 0.745 0.158 0.084 0.558 0.332 0.247 0.785 0.630

0.800 0.209 0.065 0.599 0.900 0.431 0.760 0.196 0.868 0.188 0.483 0.843 0.739 0.699 0.106

0.669 0.223 0.705 0.040 0.307 0.699 0.575 0.336 0.049 0.791 0.435 0.768 0.911 0.753 0.052

0.195 0.881 0.862 0.247 0.163 0.970 0.851 0.336 0.746 0.001 0.514 0.635 0.064 0.781 0.350

0.306 0.951 0.141 0.778 0.541 0.668 0.651 0.631 0.960 0.498 0.095 0.339 0.543 0.848 0.153

0.975 0.401 0.636 0.675 0.648 0.031 0.107 0.036 0.461 0.809 0.919 0.145 0.435 0.620 0.333

0.201 0.088 0.019 0.301 0.838 0.287 0.214 0.965 0.189 0.530 0.627 0.918 0.364 0.690 0.709

0.985 0.769 0.866 0.181 0.803 0.845 0.571 0.791 0.367 0.163 0.372 0.254 0.172 0.541 0.340

0.274 0.264 0.770 0.634 0.736 0.429 0.132 0.706 0.652 0.466 0.427 0.377 0.119 0.075 0.942

0.668 0.591 0.949 0.904 0.819 0.087 0.688 0.195 0.159 0.976 0.155 0.320 0.314 0.894 0.659

0.045 0.253 0.730 0.410 0.521 0.696 0.504 0.235 0.569 0.935 0.621 0.215 0.202 0.419 0.178

0.004 0.479 0.726 0.722 0.616 0.536 0.880 0.859 0.334 0.994 0.088 0.845 0.528 0.513 0.359

0.896 0.401 0.709 0.741 0.017 0.246 0.758 0.063 0.755 0.133 0.574 0.829 0.751 0.908 0.228

0.081 0.466 0.824 0.237 0.461 0.214 0.669 0.084 0.948 0.468 0.164 0.785 0.822 0.200 0.991

0.109 0.986 0.673 0.674 0.805 0.388 0.900 0.342 0.674 0.403 0.250 0.113 0.924 0.851 0.860

0.518 0.032 0.881 0.880 0.989 0.917 0.497 0.268 0.121 0.474 0.441 0.998 0.857 0.201 0.649

PASCAL

FERMAT

Investments2

STATISTICAL REVIEW

Overview of Applied Statistics for Financial Analysis

Antonio J. Macias

Antonio Macias

Three-step approach Capital Allocation*

Investments3

Parameters: Estimate

• expected returns

• standard deviations of returns, and

• correlations between assets

One-fund theorem: Find optimal risky portfolio or tangency portfolio

• stocks

• bonds

Two-fund theorem:According to your risk aversion, combine

• optimal risky portfolio with

• the risk-free asset (T-bills) rf

p

E(rp)

p

1-

2-

3-

* This is a very important summary slide of

what we will study in the Investment I course

rf

p

E(rp)

rrisky-assets

1. Stock trading “How-to?”

2. Risk and Return “What to measure”

Quantify parameters

3. Optimal asset allocation “Where? / Which assets? /

How much?”

4. Asset Pricing Models “How to asses?”

5. Security Valuation “Why this asset?” Later

HWSTG

Ch. 1 & 2

Antonio Macias

Learning objectives What is a random variable? How to calculate means and standard deviations? What is covariance and how to measure it?

What is correlation?

What is the mean and standard deviation of a portfolio of individual returns?

You saw these concepts in Ch. 13 of Block-Hirt, in the required course: FINA 30153

review if needed

1-Useful summary formulas2- In-Class Example

Investments4

Antonio Macias

Definition of Investment

“Current commitment of money or other resources in the expectation of reaping future benefits”

Cost – Benefit tradeoff

Stock Return:

What we pay vs. what we get

Investments5

Antonio Macias

Definition

Net return

Pt : price today

Pt+1 : price tomorrow

Dt+1 : dividend tomorrow

Investments6

1111

t

ttt

P

DPr

Antonio Macias

Example

Buy share at $50;

At end of year it is worth $55, and

Pays $2 dividend

Gross Return = (2+55)/50 =1 .14 =114%

Net Return = (2+55)/50 –1 =0 .14 =14%

Investments7

Antonio Macias

Net return

Income yield : cash payout

Capital gain/loss : change in security price

Investments8

gain/loss capitalyield income

1

11

111

t

tt

t

t

t

ttt

P

PP

P

D

P

DPr

Antonio Macias

Example (contd..)

Buy share at $50

At end of year it is worth $55, and

Pays $2 dividend

Income yield = $2/$50= 4%

Capital gain = ($55-$50)/$50=10%

Investments9

Antonio Macias

Statistics: Basics

Random Variable

Uncertain outcomes

Probability Distribution (Histogram)

List of values with their associated probability

Continuous versus Discrete

Example: Roll of a die

Investments10

X 1 2 3 4 5 6

Prob 1/6 1/6 1/6 1/6 1/6 1/6

Antonio Macias

Descriptive Statistics

Mean : Average value

Variance (standard deviation) : dispersion

Normal Distribution

Investments11

mean

s.d. s.d.

Antonio Macias

Expected (Mean) return

Investments12

where

rs : return if a state occurs

ps : probability of a state

where

rt : return in period t

pt : probability of period t is 1/T

S

s

ssrpr1

]~E[T

t

trT

r1

1]~E[

Antonio Macias

Example – mean

Mean =

=

Does not have to be one of the outcomes

Investments13

X 1 2 3 4 5 6

Prob 1/6 1/6 1/6 1/6 1/6 1/6

Antonio Macias

Mean =1/6*1 + 1/6*2 + 1/6*3 + 1/6*4 + 1/6*5 + 1/6*6

=3.5

Does not have to be one of the outcomes

Investments14

X 1 2 3 4 5 6

Prob 1/6 1/6 1/6 1/6 1/6 1/6

Example – mean

Antonio Macias

Variance

or

Shortcut:

Investments15

T

t

t )r(rT

r1

2]~E[1

]~V[

S

s

ss )r(rpr1

2]~E[]~V[

T

t

t rrT

r1

22]~E[

1]~V[

Antonio Macias

Standard deviation

Standard deviation is the square root of variance

Investments16

]~V[]~[SD rr

Antonio Macias

Example – variance

Variance

=

=

Investments17

X 1 2 3 4 5 6

Prob 1/6 1/6 1/6 1/6 1/6 1/6S

s

ss )r(rpr1

2]~E[]~V[

Antonio Macias

Variance =1/6*(1-3.5)^2 + 1/6*(2 -3.5)^2 + 1/6*(3 -3.5)^2 +

1/6*(4 -3.5)^2 + 1/6*(5 -3.5)^2 + 1/6*(6 -3.5)^2

=2.92

Investments18

X 1 2 3 4 5 6

Prob 1/6 1/6 1/6 1/6 1/6 1/6S

s

ss )r(rpr1

2]~E[]~V[

Example – variance

Antonio Macias

Example – variance (w/shortcut)

Variance

=

=

=

Investments19

X 1 2 3 4 5 6

Prob 1/6 1/6 1/6 1/6 1/6 1/6

T

t

t rrT

r1

22]~E[

1]~V[

Antonio Macias

Variance

=[1/6*1^2 + 1/6*2^2 + 1/6*3^2 + 1/6*4^2 + 1/6*5^2 + 1/6*6^2] – 3.5^2

=15.17-12.25

=2.92

Investments20

X 1 2 3 4 5 6

Prob 1/6 1/6 1/6 1/6 1/6 1/6

T

t

t rrT

r1

22]~E[

1]~V[

Example – variance (w/shortcut)

Antonio Macias

Example – standard deviation

Standard deviation

=

=

Investments21

X 1 2 3 4 5 6

Prob 1/6 1/6 1/6 1/6 1/6 1/6

Antonio Macias

Standard deviation

= 2.92

=1.71

Investments22

X 1 2 3 4 5 6

Prob 1/6 1/6 1/6 1/6 1/6 1/6

Example – standard deviation

Antonio Macias

Covariance of returns

Measure of movement in tandem

Correlation

Investments23

S

s

sss

S

s

sss

rrrrp

)r)(rr(rprr

1

21,2,1

1

2,21,121

]~E[]~E[

]~E[]~E[]~,~C[

]~[SD]~[SD

]~,~[C

21

2112

rr

rr

Antonio Macias

Correlations

Investments24

Correlation=-0.9

-2

-1

0

1

2

-2 -1 0 1 2

Correlation=+0.9

-2

-1

0

1

2

-2 -1 0 1 2

Correlation=0.0

-2

-1

0

1

2

-2 -1 0 1 2

Antonio Macias

Covariance of returns (contd..)

Another way

By the way

Correlation lies between –1.0 and 1.0

Investments25

]~[SD]~[SD]~,~[C 211221 rrrr

]~[V]~,~[C 111 rrr

Antonio Macias

Notational convention (confusion!)

Mean E[r] or or

Variance V[r] or 2

Standard deviation SD[r] or

Covariance of r1 and r2

C[r1 , r2] or 12

Correlation of r1 and r2

or 12

Investments26

211212

r

Antonio Macias

Larger example

Two stocks, 3 states of the world

Investments27

State 1 State 2 State 3

Probability 0.33 0.33 0.33

Stock A returns 15% 5% -10%

Stock B returns 10% 10% -5%

Antonio Macias

Example - means

Mean returns

Investments28

B

A

r

r

Antonio Macias

Mean returns

Investments29

%0.5%)5(3

1%10

3

1%10

3

1

%33.3%)10(3

1%5

3

1%15

3

1

B

A

r

r

Example - means

Antonio Macias

Example – Variances

Variances

Important point to remember!

Variances are not expressed in percentages

Variance of stock A is NOT x.xx%

Note that the variance has squared units = %^2!

Investments30

2

2

B

A

Antonio Macias

Variances

Important point to remember!

Variances are not expressed in percentages

Variance of stock A is NOT 1.06%

Note that the variance has squared units = %^2!

Investments

005.0%5%53

1%5%10

3

1%5%10

3

1

0106.0%3.3%103

1%3.3%5

3

1%3.3%15

3

1

2222

2222

B

A

Example – Variances

Antonio Macias

Example – Standard deviations

Standard deviations

Standard deviations are percentages

Investments32

B

A

Antonio Macias

Standard deviations

Standard deviations are percentages

Investments33

%1.7005.0

%3.100106.0

B

A

Example – Standard deviations

Antonio Macias

Example - Correlation

Covariance

Correlation

Investments34

AB

BA

ABAB

S

s

sss

S

s

sss

rrrrp

)r)(rr(rprr

1

21,2,1

1

2,21,121

]~E[]~E[

]~E[]~E[]~,~C[

Antonio Macias

Covariance

Correlation

Investments35

0067.0

%5%5%3.3%103

1

%5%10%3.3%53

1%5%10%3.3%15

3

1AB

92.00707.01027.0

0067.0

BA

ABAB

Example - Correlation

Antonio Macias

Mean and variance of sumof returns

Mean of sum is the sum of individual means

Variance of sum is NOT the sum of individual variances

Similarly, standard deviation of sum not the sum of individual standard deviations

Investments36

]~,~[C2]~[V]~[V]~~[V 212121 rrrrrr

]~[E]~[E]~~[E 2121 rrrr

Antonio Macias

Mean and variance of average of returns

Mean of average is the average of individual means

Variance of average is NOT the average of individual variances

Investments37

]~,~[C2]~[V]~[V4

1

2

~~V 2121

21 rrrrrr

]~[E]~[E2

1

2

~~E 21

21 rrrr

Antonio Macias

Example

Equal weighted portfolio of same stocks as in previous example

Estimate mean and standard deviation1. As if the portfolio were a single financial assets (for

ex. An ETF)2. Using the portfolio formulas based on the weights of

the individual assets contained in the portfolio

Investments38

State 1 State 2 State 3

Probability 0.33 0.33 0.33

Stock A returns 15% 5% -10%

Stock B returns 10% 10% -5%

Portfolio returns 12.5% 7.5% -7.5%

Antonio Macias

1- Example – portfolio mean and variance:Estimated as if portfolio were a single financial asset (imagine an ETF share)

Mean

Variance

Investments39

2BA

P

rr

r

2

P

n

i

ii RERp1

22))((σ

n

i

ii RpRE1

)(

Antonio Macias

Mean

Variance

Investments40

%2.42%5%33.32

%2.4%)5.7(3

1%5.7

3

1%5.12

3

1

BA

P

rr

r

0072.0

%2.4%5.73

1

%2.4%5.73

1%2.4%5.12

3

1

2

222

P

1- Example – portfolio mean and varianceEstimated as if portfolio were a single financial asset (imagine an ETF share)

Antonio Macias

Mean

Variance

Investments41

2- Example – portfolio mean and variance using weights of assets in portfolio

)()(1

j

m

j

jp REwRE

ijj

m

ji

iwwVariance1,

2

Antonio Macias

Mean

Mean with 2 assets in the portfolio:

Investments42

2-Example – portfolio mean using weights of assets in portfolio

)()(1

j

m

j

jp REwRE

)()()( 2211 REwREwRE p

Antonio Macias

Variance

Variance with m=2 assets in the portfolio:

Investments43

2-Example – portfolio variance using weights of assets in portfolio

ijj

m

ji

iwwVariance1,

2

122211

2

2

2

2

2

1

2

1

1221

2

2

2

2

2

1

2

1

2222211212211111

2

))((2

2

wwww

wwww

wwwwwwwwVariance

Where 12 is the covariance between asset 1 and 2

And 12 is the correlation between asset 1 and 2

Antonio Macias

2-Example – portfolio standard deviation:based on 2 assets with weight_A = weight_B = 1/2

Standard deviation

Alternatively to first estimation method, we could have used portfolio formulas (2nd

method)*

* It is sometimes easier to apply the more general formula as if we are estimating the std. deviation of a single asset (in this case, the portfolio0, since it is more general

Investments44

2BA

P

ABBA

ABBAP

24

1

2

1*

2

1*2

2

1

2

1

22

2

2

2

2

2

ijj

m

ji

iwwVariance1,

2

Antonio Macias

Standard deviation

Using 2nd method based on portfolio formulas

Investments45

2

%5.80072.0

BA

P

00725.0

0.0067*20.0050.01064

1

24

1 222

ABBAP

Example – portfolio standard deviation

Estimated based on 1st

method as if portfolio

were a single financial

asset

ijj

m

ji

iwwVariance1,

2

Antonio Macias

Investments46

Suppose you have predicted the following returns for stocks C and T in three possible states of nature. What are the expected returns, E[r]? And the standard deviations, ?

State Probability C T .Boom 0.3 0.15 0.25Normal 0.5 0.10 0.20Recession ??? 0.02 0.01

E[rC]= C=E[rT]= T=

What would be the portfolio return, E[rportfolio], and its standard deviation, portfolio, if the weights of the individual assets are:wC = 0.4 and wT = 0.6

In Class Example – Individual and Portfolio returns and standard deviation

Antonio Macias

Investments47

Summary of Useful Formulas:

)()(1

i

N

i

ip rEwrE

ijj

N

j

i

N

i

wwVariance11

2

ji

ij

ijijrhojincorrelatio ),(

s

M

s

si rprE1

)(

2

1

2 ])[(* i

M

s

ss rErp

1111

t

ttt

P

DPr

VarianceDevStd 2..

Stock Return

Portfolio levelIndividual asset level

Where s: state of nature; p: probability of state s; M: number of states of nature;

i: asset i; j: asset j; N: number of assets in portfolio

Antonio Macias

Investments48

1. Expected Value of x:

where x is the random variable, and pi is the probability of state i.

2.A. Expected return of a Portfolio, given probabilities for n different states of nature and the return, Ri for each state of nature

(“branch”)

3. Variance of expected observations:

where x is the random variable, and pi is the probability of state i.

4. Portfolio estimations (given all the j assets included in the portfolio)

4.A Mean-return of a Portfolio:

4.B Variance of a Portfolio

4.C Correlation coefficient between two assets:

Where: wi is the weight on asset i with a total of m assets in the portfolio.

ij is the covariance between asset i and j

ij is the correlation coefficient between asset i and j

5. Covariance of expected Observations:

where XAi, X Bi are the values of random variables

A and B in state i. p(si) is the probability of state i.

Various Useful detailed Formulas

i

N

i

i pxxE1

][

i

n

i

i RpRE1

)(

i

N

i

i pxEx *])[( 2

1

2

)()(1

j

m

j

jp REwREijj

m

ji

iwwVariance1,

2

:

ji

ij

ijijrho

states. Nfor

])[])([()( ...

])[])([()(

])[])([()(),(C

222

111

BBNAANN

BBAA

BBAABA

XEXXEXsp

XEXXEXsp

XEXXEXspXXOV

Note that the letter you use for each index and for the total number of elements

for each summation can change, as long as you are consistent with your

problem at hand. The nomenclature is just and aid to solve the problems.

Antonio Macias

Investments49

1-Summation, or Sigma, Notation

http://www.math.montana.edu/frankw/ccp/general/sigma/learn.htm

Great place to find an introduction to the topic, with many

Applied examples to assess your understanding

2- Series (mathematics)

http://en.wikipedia.org/wiki/Series_(mathematics)

From Wikipedia, the free encyclopedia

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last

terms, whereas infinite sequences and series continue indefinitely. [1]

In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of

adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more compactly using

the summation symbol ∑. An example is the famous series from Zeno's dichotomy

The terms of the series are often produced according to a certain rule, such as by a formula, or by

an algorithm. As there are an infinite number of terms, this notion is often called an infinite series.

Unlike finite summations, infinite series need tools from mathematical analysis to be fully understood

and manipulated. In addition to their ubiquity in mathematics, infinite series are also widely used in

other quantitative disciplines such as physics and computer science.

Appendix: Useful Resources about Summations (Series)

:

Antonio Macias

Investments50

Statistical and Probability Overview

Key Concepts for Investment Management: What is a random variable? How to calculate means and standard deviation? What is covariance and how to measure it? What is correlation?

What is the mean and standard deviation of a portfolio of individual returns?

Skim Chapter 5 for application in Finance (we will see more later)

Online resources: http://www.uidaho.edu/stat/scc/review1.htm http://homepages.wmich.edu/~bwagner/StatReview/index.html http://hspm.sph.sc.edu/COURSES/J716/a01/stat.html

Antonio Macias

Three-step approach Capital Allocation*

Investments51

Parameters: Estimate

• expected returns

• standard deviations of returns, and

• correlations between assets

One-fund theorem: Find optimal risky portfolio or tangency portfolio

• stocks

• bonds

Two-fund theorem:According to your risk aversion, combine

• optimal risky portfolio with

• the risk-free asset (T-bills) rf

p

E(rp)

p

1-

2-

3-

* This is a very important summary slide of

what we will study in the Investment I course

rf

p

E(rp)

rrisky-assets

1. Stock trading “How-to?”

2. Risk and Return “What to measure”

Quantify parameters

3. Optimal asset allocation “Where? / Which assets? /

How much?”

4. Asset Pricing Models “How to asses?”

5. Security Valuation “Why this asset?” Later

HWSTG

Ch. 1-3

Ch. 6

Ch. 5

• Ch. 7&on

• *HW7

DCF